Formel von Cardano (CAS)

exakte Lösung von allgemeinen Gleichungen dritten Grades
Aus wxMaxima
[br][math] [br]x=\left( -\frac{\sqrt{3}\,i}{2}-\frac{1}{2}\right) \,{\left( \frac{\sqrt{27\,{a}^{2}\,{d}^{2}+\left( 4\,{b}^{3}-18\,a\,b\,c\right) \,d+4\,a\,{c}^{3}-{b}^{2}\,{c}^{2}}}{2\,{3}^{\frac{3}{2}}\,{a}^{2}}-\frac{27\,{a}^{2}\,d-9\,a\,b\,c+2\,{b}^{3}}{54\,{a}^{3}}\right) }^{\frac{1}{3}}-\frac{\left( \frac{\sqrt{3}\,i}{2}-\frac{1}{2}\right) \,\left( 3\,a\,c-{b}^{2}\right) }{9\,{a}^{2}\,{\left( \frac{\sqrt{27\,{a}^{2}\,{d}^{2}+\left( 4\,{b}^{3}-18\,a\,b\,c\right) \,d+4\,a\,{c}^{3}-{b}^{2}\,{c}^{2}}}{2\,{3}^{\frac{3}{2}}\,{a}^{2}}-\frac{27\,{a}^{2}\,d-9\,a\,b\,c+2\,{b}^{3}}{54\,{a}^{3}}\right) }^{\frac{1}{3}}}-\frac{b}{3\,a},\\[br]x=\left( \frac{\sqrt{3}\,i}{2}-\frac{1}{2}\right) \,{\left( \frac{\sqrt{27\,{a}^{2}\,{d}^{2}+\left( 4\,{b}^{3}-18\,a\,b\,c\right) \,d+4\,a\,{c}^{3}-{b}^{2}\,{c}^{2}}}{2\,{3}^{\frac{3}{2}}\,{a}^{2}}-\frac{27\,{a}^{2}\,d-9\,a\,b\,c+2\,{b}^{3}}{54\,{a}^{3}}\right) }^{\frac{1}{3}}-\frac{\left( -\frac{\sqrt{3}\,i}{2}-\frac{1}{2}\right) \,\left( 3\,a\,c-{b}^{2}\right) }{9\,{a}^{2}\,{\left( \frac{\sqrt{27\,{a}^{2}\,{d}^{2}+\left( 4\,{b}^{3}-18\,a\,b\,c\right) \,d+4\,a\,{c}^{3}-{b}^{2}\,{c}^{2}}}{2\,{3}^{\frac{3}{2}}\,{a}^{2}}-\frac{27\,{a}^{2}\,d-9\,a\,b\,c+2\,{b}^{3}}{54\,{a}^{3}}\right) }^{\frac{1}{3}}}-\frac{b}{3\,a},\\[br]x={\left( \frac{\sqrt{27\,{a}^{2}\,{d}^{2}+\left( 4\,{b}^{3}-18\,a\,b\,c\right) \,d+4\,a\,{c}^{3}-{b}^{2}\,{c}^{2}}}{2\,{3}^{\frac{3}{2}}\,{a}^{2}}-\frac{27\,{a}^{2}\,d-9\,a\,b\,c+2\,{b}^{3}}{54\,{a}^{3}}\right) }^{\frac{1}{3}}-\frac{3\,a\,c-{b}^{2}}{9\,{a}^{2}\,{\left( \frac{\sqrt{27\,{a}^{2}\,{d}^{2}+\left( 4\,{b}^{3}-18\,a\,b\,c\right) \,d+4\,a\,{c}^{3}-{b}^{2}\,{c}^{2}}}{2\,{3}^{\frac{3}{2}}\,{a}^{2}}-\frac{27\,{a}^{2}\,d-9\,a\,b\,c+2\,{b}^{3}}{54\,{a}^{3}}\right) }^{\frac{1}{3}}} [/math]

Information: Formel von Cardano (CAS)